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We give a formula for the inverse matrix to an infinite matrix with possibly noncommutative entries, generalizing the Newton interpolation formula and the Taylor formula.
Consider a symmetric matrix $A(v)\in\RR^{n\times n}$ depending on a vector $v\in\RR^n$ and satisfying the property $A(\alpha v)=A(v)$ for any $\alpha\in\RR\backslash{0}$. We will here study the problem of finding $(\lambda,v)\in\RR\times…
Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as…
This article develops a novel approach to the representation of singular integral operators of Calder\'on-Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is…
By using a real matrix translation, we propose a coupled eigenvalue problem for octonionic operators. In view of possible applications in quantum mechanics, we also discuss the hermiticity of such operators. Previous difficulties in…
Now a final and maybe simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a {\it spacial domain with an outer boundary} over a finite time interval is fixed. The…
A new exponentially convergent algorithm is proposed for an abstract the first order differential equation with unbounded operator coefficient possessing a variable domain. The algorithm is based on a generalization of the Duhamel integral…
Computing many eigenpairs of the Schr{\"o}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a…
We present an effective operator formalism for open quantum systems. Employing perturbation theory and adiabatic elimination of excited states for a weakly driven system, we derive an effective master equation which reduces the evolution to…
Presented here is a matrix inversion method utilizing quantum searching algorithm. In this method, huge Hilbert space as a whole spanned by myriad of eigen states is searched and evaluated efficiently by sequential reduction in dimension…
This note focuses on the direct spectral problem for canonical Hamiltonian systems on the half-line $\mathbb{R}_+$. Truncated Toeplitz operators have been effectively used to solve the inverse spectral problem when the spectral measure is a…
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a $c$-number function and a power series of perturbation including all…
Four new examples of explicitly diagonalizable Hankel matrices depending on a parameter $k\in(0,1)$ are presented. The Hankel matrices are regarded as matrix operators on the Hilbert space $\ell^{2}(\mathbb{N}_{0})$ and the solution of the…
We investigate bicomplex Hamiltonian systems in the framework of an analogous version of the Schrodinger equation. Since in such a setting three different types of conjugates of bicomplex numbers appear, each is found to define in a natural…
Brief introduction to the discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation…
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient…
In this paper we show how to calculate explicitly the exponential of certain matrices, which are evolution operators governing the interaction of the four level system of atoms and the radiation, etc. We present a consistent method in terms…
Using the Hubbard representation for $SU(2)$ we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of non-linear coupled equations. In order to…
This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…